## 3.3 Single population proportion

- The population parameter \(p\) of the Binomial distribution can also be tested using similar procedure

\[\begin{equation} H_0:~~p=p_0 \tag{3.9} \end{equation}\]

Assumed population proportion \(p_0\) is the value that we think is true. If sample data shows that this is false, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis.

Three alternative hypothesis are possible:

\[\begin{align} H_1:&~~p \ne p_0 &\text{two-tailed test} \\ \\ H_1:&~~p < p_0 &\text{left-tailed test} \\ \\ H_1:&~~p > p_0 &\text{right-tailed test} \\ \tag{3.10} \end{align}\]

- As the binomial distribution of a sample proportion \(\hat{p}\) can be approximated by the normal distribution, the test statistic follows a standard normal distribution:

\[\begin{align} Z&=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} \\ \\ \hat{p}&=\frac{x}{n} \\ \tag{3.11} \end{align}\]

**Example 3.5 **Suppose a consumer group suspects that the proportion of households that have three or more cell phones is \(30\%\). A cell phone company has reason to believe that the proportion is not \(30\%\). Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey \(150\) households with the result that \(43\) of them have three or more cell phones. Level of significance is \(5%\).

**Example 3.6 **Test the hypothesis that more than \(50\%\) of the companies are listed on the stock exchange. Level of significance is \(5\%\).